cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A166575 Primes p>=5 with the property: if Prime(k)

=Prime(k)+ Prime(k+1).

Original entry on oeis.org

5, 13, 19, 31, 37, 43, 53, 61, 71, 73, 79, 101, 103, 113, 131, 139, 157, 163, 173, 191, 193, 199, 211, 223, 241, 251, 269, 271, 277, 293, 311, 313, 331, 353, 373, 379, 397, 419, 421, 439, 443, 457, 463, 499, 509, 521, 523, 541, 577, 601, 607, 613, 619, 631, 653, 659, 661, 673, 691
Offset: 1

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Author

Vladimir Shevelev, Oct 17 2009

Keywords

Comments

If A(x) is the counting function of a(n) not exceeding x, then, in view of the symmetry, it is natural to conjecture that A(x)~pi(x)/2.

Examples

			Let p=13. Then we have 5<13/2<7. Since 13>5+7, then 13 is in the sequence.

Crossrefs

Cf. A166574, A182365, A166307, A166252, A166251, A164368, A104272, A080359, A164333, A164288, A164294, A164554

Programs

  • Mathematica
    Reap[Do[p=Prime[n]; k=PrimePi[p/2]; If[p>=Prime[k]+Prime[k+1], Sow[p]], {n,3,PrimePi[1000]}]][[2,1]]

A182451 Numbers of A166252 which are not in A164554.

Original entry on oeis.org

109, 151, 191, 229, 233, 283, 311, 571, 643, 683, 727, 941, 991, 1033, 1051, 1373, 1493, 1667, 1697, 1741, 1747, 1783, 1787, 1801, 1931
Offset: 1

Views

Author

Vladimir Shevelev, Apr 29 2012

Keywords

Comments

All Ramanujan primes (A104272) are in A164368 and all Labos primes (A080359) are in A194598. Peculiar primes (see comment in A164554)are simultaneously Ramanujan and Labos primes, while central primes (A166252) are in the intersection of A164368 and A194598 for n>=2. Hence, for n>=2, all peculiar primes are central primes, but conversely is not true. The sequence lists non-peculiar central numbers.

Crossrefs

Cf. A164554, A166252, A104272, A080359, A164368, A194598.

A352127 a(n) is the smallest even number whose n smallest divisors > 2 are all odd.

Original entry on oeis.org

2, 6, 30, 2002, 92378, 13357342, 2697562774, 292157776958, 36257787561098, 5563815981553006, 406158566653369438, 757411721425002359054, 78013407306775242982562, 17167365460734333186898502, 1939912297062979650119530726, 2548800423984304257055702381202
Offset: 0

Views

Author

Jon E. Schoenfield, Mar 05 2022

Keywords

Comments

For n > 1, if the set of divisors of an even number m begins with 1, 2, and n odd divisors, then m must be divisible by 2 but not by 4, and its smallest odd divisor > 1 must be a prime p such that m has at least n odd divisors in the interval [p, 2*p-1] (since 2*p will be an even divisor), all of which must be prime (since, if any were composite, then p would not be m's smallest divisor > 1). Thus, the smallest such m is twice the product of the first run of n consecutive primes, the largest and smallest of which have a ratio less than 2.

Examples

			The terms and their factorizations begin:
  a(0)  =                  2 = 2
  a(1)  =                  6 = 2 *  3
  a(2)  =                 30 = 2 *  3* 5
  a(3)  =               2002 = 2 *  7*11*13
  a(4)  =              92378 = 2 * 11*13*17*19
  a(5)  =           13357342 = 2 * 17*19*23*29*31
  a(6)  =         2697562774 = 2 * 23*29*31*37*41*43
  a(7)  =       292157776958 = 2 * 29*31*37*41*43*47*53
  a(8)  =     36257787561098 = 2 * 31*37*41*43*47*53*59*61
  a(9)  =   5563815981553006 = 2 * 37*41*43*47*53*59*61*67*71
  a(10) = 406158566653369438 = 2 * 37*41*43*47*53*59*61*67*71*73
  ...
a(n-1) divides a(n) whenever a(n-1) and a(n) have the same smallest odd prime factor; this happens at n = 2, 10, 12, 14, 19, 20, ..., which are the indices at which the largest prime factor of a(n) is less than twice the smallest prime factor of a(n-1). E.g., both a(9) and a(10) have 37 as their smallest odd prime factor, and 73 (the largest prime factor of a(10)) < 74 = 2*37.

Crossrefs

Cf. A080359.

Formula

a(n) = 2*Product_{j=k+1..k+n} prime(j) where k is the smallest integer such that 2*prime(k+1) > prime(k+n).
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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